Moving frames on the twistor space of self-dual positive Einstein 4-manifolds

TL;DR

This paper compares two families of Riemannian metrics on the twistor space of self-dual positive Einstein 4-manifolds, analyzing their Ricci tensors and Ricci flow behavior to provide a new proof of their classification.

Contribution

It introduces a novel comparison between canonical deformation metrics and Chow-Yang metrics on twistor spaces, offering new insights into their geometric properties.

Findings

The Ricci tensors of the two metric families are explicitly compared.

Under Ricci flow, the behaviors of the two families are characterized.

A new proof is provided for the classification of self-dual positive Einstein 4-manifolds.

Abstract

The twistor space of self-dual positive Einstein manifolds naturally admits two 1-parameter families of Riemannian metrics, one is the family of canonical deformation metrics and the other is the family introduced by B. Chow and D. Yang in 1989. The purpose of this paper is to compare these two families. In particular we compare the Ricci tensor and the behavior under the Ricci flow of these families. As an application, we propose a new proof to the fact that a locally irreducible self-dual positive Einstein 4-manifold is isometric to either $S^4$ with a standard metric or $\PP^2(\C)$ with a Fubini-Study metric.